6.853: Topics in Algorithmic Game Theory Fall 2011 Constantinos Daskalakis Lecture 11

Last Lecture... 0n0n Generic PPAD Embed PPAD graph in [0,1] 3 3D-SPERNER canonical p.w. linear BROUWER multi-player NASH 4-player NASH 3-player NASH 2-player NASH [Pap ’94] [DGP ’05] [DP ’05] [CD’05] [CD’06] DGP = Daskalakis, Goldberg, Papadimitriou CD = Chen, Deng

This Lecture... 0n0n Generic PPAD Embed PPAD graph in [0,1] 3 3D-SPERNER multi-player NASH 4-player NASH 3-player NASH 2-player NASH [Pap ’94] [DGP ’05] [DP ’05] [CD’05] [CD’06] DGP = Daskalakis, Goldberg, Papadimitriou CD = Chen, Deng canonical p.w. linear BROUWER

Polymatrix Games (Review) The PPAD-hard games we constructed were separable multiplayer games, aka polymatrix games. These are multi-player games with edge-wise separable utility functions. - player’s payoff is the sum of payoffs from all adjacent edges … … - edges are 2-player games

Reducing Polymatrix to Bimatrix Games … … polymatrix game w.l.o.g. can assume bipartite, by turning every gadget used in the reduction into a bipartite game (inputs&output are on one side and “middle player” is on the other side)

… … 2-player game polymatrix game red lawyer represents red nodes, while blue lawyer represents blue nodes w.l.o.g. can assume bipartite, by turning every gadget used in the reduction into a bipartite game (inputs&output are on one side and “middle player” is on the other side) Reducing Polymatrix to Bimatrix Games

Payoffs of the Lawyer-Game - wishful thinking: if (x, y) is a Nash equilibrium of the lawyer-game, then the marginal distributions that x assigns to the strategies of the red nodes and the marginals that y assigns to the blue nodes, comprise a Nash equilibrium. But why would a lawyer play every node he represents? … …

Enforcing Fairness - The lawyers play on the side a high-stakes game. - W.l.o.g. assume that each lawyer represents n clients. Name these clients 1,…,n. - Payoffs of the high-stakes game: Suppose the red lawyer plays any strategy of client j, and blue lawyer plays any strategy of client k, then = M If, then red lawyer gets +M, while blue lawyer gets –M. If, then both players get 0.

Enforcing Fairness Claim: The unique Nash equilibrium of the high-stakes lawyer game is for both lawyers to play uniformly over their clients. Proof:exercise

Enforcing Fairness + M,-M0,0 M,-M0, 0 M,-M M = high stakes game payoff table addition Choose:

Analyzing the Lawyer Game - when it comes to distributing the total probability mass among the different nodes of, essentially only the high-stakes game is relevant to the lawyers… Lemma 1: if (x, y) is an equilibrium of the lawyer game, for all u, v : - when it comes to distributing the probability mass x u among the different strategies of node u, only the payoffs of the game are relevant… The payoff difference for the red lawyer from strategies and is Lemma 2: Proof:exercise total probability mass assigned by lawyers on nodes u, v respectively

Analyzing the Lawyer Game (cont.) Lemma 2  if, then for all j: - if M is large, can correct it to an exact Nash equilibrium of the polymatrix game, using similar technique as exercise of last time. (marginals given by lawyers to different nodes) - define and Observation: if we had x u =1/n, for all u, and y v =1/n, for all v, then would be a Nash equilibrium. - the deviation from uniformity results in an approximate Nash equilibrium of the polymatrix game.

lawyer construction Exercise form Last time Exercise from Last Time obvious through SPERNER, BROUWER

Algorithms for Nash Equilibria Simplicial Approximation Algorithms Support Enumeration Algorithms Lipton-Markakis-Mehta Algorithms for Symmetric Games (next time)The Lemke-Howson Algorithm (next time)

Algorithms for Nash Equilibria Simplicial Approximation Algorithms

Given a continuous function, where f satisfies a Lipschitz condition and S is a compact convex subset of the Euclidean space, find such that.. (or exhibit a pair of points violating the Lipschitz condition, or a point mapped by the function outside of S) suppose that S is described in some meaningful way in the input, e.g. polytope, or ellipsoid Simplicial Approximation Algorithms comprise a family of algorithms computing an approximate fixed point of f by dividing S up into simplices and defining a walk that pivots from simplex to simplex of the subdivision until it settles at a simplex containing an approximate fixed point. (this is a re-iteration of the BROUWER problem that we defined in earlier lectures; for details on how to make the statement formal check previous lectures)

(our own) Simplicial Approximation Algorithm (details in Lecture 6,7) 1. Embed S into a large enough hypercube. 2. Define an extension f ’ of f to the points in the hypercube that lie outside of S in a way that, given an approximate fixed point of f ’, an approximate fixed point of f can be obtained in polynomial time. 3. Define the canonical subdivision of the hypercube (with small enough precision that depends on the Lipschitz property of f ’ see previous lectures ). 4. Color the vertices of the subdivision with n +1 colors, where n is the dimensionality of the hypercube. The color at a point x corresponds to the angle of the displacement vector. 5. The colors define a legal Sperner coloring. 6. Solve the Sperner instance, by defining a directed walk starting at the “starting simplex” (defined in lecture 6) and pivoting between simplices through colorful facets. 7. One of the corners of the simplex where the walk settles is an approximate fixed point. the non- constructive step

Algorithms for Nash Equilibria Simplicial Approximation Algorithms Support Enumeration Algorithms

How better would my life be if I knew the support of the Nash equilibrium? … and the game is 2-player? any feasible point (x, y) of the following linear program is an equilibrium! Setting: Let (R, C) be an m by n game, and suppose a friend revealed to us the supports and respectively of the Row and Column players’ mixed strategies at some equilibrium of the game. s.t. and

Support Enumeration Algorithms How better would my life be if I knew the support of the Nash equilibrium? … and the game is 2-player? for guessing the support for solving the LP Runtime:

Support Enumeration Algorithms How better would my life be if I knew the support of the Nash equilibrium? … and the game is polymatrix?  can do this with Linear Programming too! input:the support of every node at equilibrium goal:recover the Nash equilibrium with that support the idea of why this is possible is similar to the 2-player case: - the expected payoff of a node from a given pure strategy is linear in the mixed strategies of the other players; - hence, once the support is known, the equilibrium conditions correspond to linear equations and inequalities.

Rationality of Equilibria Important Observation: The correctness of the support enumeration algorithm implies that in 2- player games and in polymatrix games there always exists an equilibrium in rational numbers, and with description complexity polynomial in the description of the game!

Algorithms for Nash Equilibria Simplicial Approximation Algorithms Support Enumeration Algorithms Lipton-Markakis-Mehta

Computation of Approximate Equilibria Theorem [Lipton, Markakis, Mehta ’03]: For all and any 2-player game with at most n strategies per player and payoff entries in [0,1], there exists an -approximate Nash equilibrium in which each player’s strategy is uniform on a multiset of their pure strategies of size - By Nash’s theorem, there exists a Nash equilibrium (x, y). - Suppose we take samples from x, viewing it as a distribution. : uniform distribution over the sampled pure strategies - Similarly, define by taking t samples from y. Claim: Proof idea: (of a stronger claim)

Computation of Approximate Equilibria Lemma: With probability at least 1-4/n the following are satisfied: Proof: on the board using Chernoff bounds. Suffices to show the following:

Computation of Approximate Equilibria set : every point is a pair of mixed strategies that are uniform on a multiset of size Random sampling from takes expected time Oblivious Algorithm: set does not depend on the game we are solving. Theorem [Daskalakis-Papadimitriou ’09] : Any oblivious algorithm for general games runs in expected time